As soon as I read your post title, I thought, why, .375H&H or .338WM or 8mm mag-type. Soon as you included Brown bear, it's a definite for a good .375mag. No silly stuff like the Ultra's, just the plain old .375.
: The recoil is soft compared to other rounds in it's class and I find the .375 easier to shoot than about any classic stocked .338.
: With the newer 260gr. Nosler Partition, you can handle about everything, but 285GS and heavy partitions really seem to work well on the big bears.
: The .338 works too, but the 3/8" bore seems to have a bit more.